Weak convergence from projected laws on a positive-measure set of directions

2026-04-11

AI Summary

This study addresses the problem of inferring weak convergence of high-dimensional probability measures from convergence of their one-dimensional projections over a limited set of directions. While the classical Cramér–Wold theorem requires convergence along all directions, this work demonstrates that under moment determinacy conditions, weak convergence follows already from convergence of projections over any set of directions with positive spherical measure—such as almost every randomly chosen direction. By integrating tools from weak convergence theory, absolute continuity of spherical measures, and moment determinacy analysis, the authors establish a novel criterion for weak convergence based on positive-measure direction sets and provide an intuitive probabilistic interpretation. This result substantially relaxes the assumptions of the classical theorem while preserving its conclusion.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Other Foundations of Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

The Cram\'er--Wold device characterises weak convergence of probability measures on~$\R^d$ through convergence of all one-dimensional projected laws. We prove that, if the target projected laws are moment-determinate for surface-almost every direction, then weak convergence already follows from projected convergence on a positive-measure set of directions. This yields a simple probabilistic interpretation: if one samples a direction at random from any distribution on the sphere that is absolutely continuous with respect to surface measure, then, with probability one, convergence of the projected law along the sampled direction already forces global weak convergence under the same moment-determinacy assumption.

Problem

• weak convergence
• Cramér-Wold device
• projected laws
• moment-determinacy
• positive-measure set

Innovation

• weak convergence
• Cramér-Wold device
• moment-determinacy
• projected laws
• positive-measure directions